3. Each of J, K, L, M andN is a linear transformation from R2 to R?. These functions are given as follows: J(11, T2) = (5x1- 3r2,-101, + 6x2), K(I1, 12) = (-12, I), L(I1, T2) = (r2, T1), M(r1, 12) = (3xı + 12, I1 + 2r2), N(r1, T2) = (-1, I2). (a) In each case, write down the matrix of the transformation and compute its deter- minant. (b) Sketch a picture indicating what happens to the standard basis {(1.0), (0, 1)} un- der each function. (c) One of these functions is not injective. WVhich is it? (d) One of these functions is an anti-clockwise rotation of the plane. Which is it? (e) One of these functions is a reflection over the vertical axis. Which is it? (f) Which functions preserve and which reverse orientation?

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Question 3.

3. Each of J, K, L, M and N is a linear transformation from R2 to R?. These functions
are given as follows:
J(11, 12) = (5x1- 3r2,-10r1 + 6x2),
K(I1, 12) = (-r2, T1),
L(r1, 12) = (r2, x1),
M(11, 12) = (3xı+ 12, T1 + 2r2),
N(11, 12) = (-I1, T2).
%3D
%3D
(a) In each case, write down the matrix of the transformation and compute its deter-
minant.
(b) Sketch a picture indicating what happens to the standard basis {(1.0), (0. 1)} un-
der each function.
(c) One of these functions is not injective. Which is it?
(d) One of these functions is an anti-clockwise rotation of the plane. Which is it?
(e) One of these functions is a reflection over the vertical axis. Which is it?
(f) Which functions preserve and which reverse orientation?
Transcribed Image Text:3. Each of J, K, L, M and N is a linear transformation from R2 to R?. These functions are given as follows: J(11, 12) = (5x1- 3r2,-10r1 + 6x2), K(I1, 12) = (-r2, T1), L(r1, 12) = (r2, x1), M(11, 12) = (3xı+ 12, T1 + 2r2), N(11, 12) = (-I1, T2). %3D %3D (a) In each case, write down the matrix of the transformation and compute its deter- minant. (b) Sketch a picture indicating what happens to the standard basis {(1.0), (0. 1)} un- der each function. (c) One of these functions is not injective. Which is it? (d) One of these functions is an anti-clockwise rotation of the plane. Which is it? (e) One of these functions is a reflection over the vertical axis. Which is it? (f) Which functions preserve and which reverse orientation?
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